The substituents R₁ and R₂ for nine peptides are listed in the table given below. How many of these peptides are positively charged at pH = 7.0 ?

Peptide	$${R_1}$$	$${R_2}$$
I	H	H
II	H	$$C{H_3}$$
III	$$C{H_2}COOH$$	H
IV	$$C{H_2}CON{H_2}$$	$${(C{H_2})_4}N{H_2}$$
V	$$C{H_2}CON{H_2}$$	$$C{H_2}CON{H_2}$$
VI	$${(C{H_2})_4}N{H_2}$$	$${(C{H_2})_4}N{H_2}$$
VII	$$C{H_2}COOH$$	$$C{H_2}CON{H_2}$$
VIII	$$C{H_2}OH$$	$${(C{H_2})_4}N{H_2}$$
IX	$${(C{H_2})_4}N{H_2}$$	$$C{H_3}$$

When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is ____________.

Let $$\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$$ be such that
$$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \left( {2\pi - \theta } \right) > 0$$ and $$ - 1 < \sin \theta \, < - {{\sqrt 3 } \over 2},$$

then $$\varphi $$ cannot satisfy

Let z be a complex number such that the imaginary part of z is non-zero and $$a\, = \,{z^2} + \,z\, + 1$$ is real. Then a cannot take the value